updating splines
This commit is contained in:
mhjensen
@@ -69,74 +69,86 @@ Automatically generated HTML file from DocOnce source
|
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('Steepest descent method', 2, None, '___sec19'),
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('Steepest descent method', 2, None, '___sec20'),
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||||
('Final expressions', 2, None, '___sec21'),
|
||||
('Code examples for steepest descent', 2, None, '___sec22'),
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('Simple codes for steepest descent and conjugate gradient '
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'using a $2\\times 2$ matrix, in c++, Python code to come',
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None,
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||||
'___sec23'),
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('The routine for the steepest descent method',
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||||
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('Conjugate gradient', 2, None, '___sec25'),
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||||
('Revisiting our first homework', 2, None, '___sec26'),
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||||
('Gradient descent example', 2, None, '___sec27'),
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||||
('The derivative of the cost/loss function', 2, None, '___sec28'),
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('The Hessian matrix', 2, None, '___sec29'),
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||||
('Simple program', 2, None, '___sec30'),
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||||
('Gradient Descent Example', 2, None, '___sec31'),
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'___sec24'),
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||||
('Steepest descent example', 2, None, '___sec25'),
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||||
('Conjugate gradient', 2, None, '___sec26'),
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||||
('Revisiting our first homework', 2, None, '___sec27'),
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||||
('Gradient descent example', 2, None, '___sec28'),
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('The derivative of the cost/loss function', 2, None, '___sec29'),
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('The Hessian matrix', 2, None, '___sec30'),
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('Simple program', 2, None, '___sec31'),
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('Gradient Descent Example', 2, None, '___sec32'),
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('And a corresponding example using _scikit-learn_',
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||||
None,
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||||
'___sec32'),
|
||||
('Gradient descent and Ridge', 2, None, '___sec33'),
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||||
('Automatic differentiation', 2, None, '___sec34'),
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||||
('Using autograd', 2, None, '___sec35'),
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('Autograd with more complicated functions', 2, None, '___sec36'),
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'___sec33'),
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||||
('Gradient descent and Ridge', 2, None, '___sec34'),
|
||||
('Automatic differentiation', 2, None, '___sec35'),
|
||||
('Using autograd', 2, None, '___sec36'),
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('Autograd with more complicated functions', 2, None, '___sec37'),
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('More complicated functions using the elements of their '
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'arguments directly',
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2,
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None,
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'___sec37'),
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'___sec38'),
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('Functions using mathematical functions from Numpy',
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2,
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None,
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'___sec38'),
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||||
('More autograd', 2, None, '___sec39'),
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||||
('And with loops', 2, None, '___sec40'),
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('Using recursion', 2, None, '___sec41'),
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('Unsupported functions', 2, None, '___sec42'),
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'___sec39'),
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('More autograd', 2, None, '___sec40'),
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('And with loops', 2, None, '___sec41'),
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('Using recursion', 2, None, '___sec42'),
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('Unsupported functions', 2, None, '___sec43'),
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('The syntax a.dot(b) when finding the dot product',
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2,
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None,
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'___sec43'),
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('Recommended to avoid', 2, None, '___sec44'),
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('Stochastic Gradient Descent', 2, None, '___sec45'),
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('Computation of gradients', 2, None, '___sec46'),
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('SGD example', 2, None, '___sec47'),
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('The gradient step', 2, None, '___sec48'),
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('Simple example code', 2, None, '___sec49'),
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||||
('When do we stop?', 2, None, '___sec50'),
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('Slightly different approach', 2, None, '___sec51'),
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('Program for stochastic gradient', 2, None, '___sec52'),
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('Momentum based methods', 2, None, '___sec53'),
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('Conjugate gradient method', 2, None, '___sec54'),
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'___sec44'),
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('Recommended to avoid', 2, None, '___sec45'),
|
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('Stochastic Gradient Descent', 2, None, '___sec46'),
|
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('Computation of gradients', 2, None, '___sec47'),
|
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('SGD example', 2, None, '___sec48'),
|
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('The gradient step', 2, None, '___sec49'),
|
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('Simple example code', 2, None, '___sec50'),
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('When do we stop?', 2, None, '___sec51'),
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('Slightly different approach', 2, None, '___sec52'),
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('Program for stochastic gradient', 2, None, '___sec53'),
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('Momentum based methods', 2, None, '___sec54'),
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('Conjugate gradient method', 2, None, '___sec55'),
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('Conjugate gradient method', 2, None, '___sec56'),
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('Conjugate gradient method', 2, None, '___sec57'),
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('Conjugate gradient method and iterations', 2, None, '___sec58'),
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('Conjugate gradient method', 2, None, '___sec59'),
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('Conjugate gradient method', 2, None, '___sec58'),
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('Conjugate gradient method and iterations', 2, None, '___sec59'),
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('Conjugate gradient method', 2, None, '___sec60'),
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('Conjugate gradient method', 2, None, '___sec61'),
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('Conjugate gradient method', 2, None, '___sec62'),
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('Simple implementation of the Conjugate gradient algorithm',
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2,
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None,
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'___sec62'),
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'___sec63'),
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('Broyden–Fletcher–Goldfarb–Shanno algorithm',
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2,
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None,
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'___sec63')]}
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'___sec64'),
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('Using gradient descent methods, limitations',
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2,
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None,
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'___sec65'),
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('Momentum based GD', 2, None, '___sec66'),
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('More on momentum based approaches', 2, None, '___sec67'),
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('Momentum parameter', 2, None, '___sec68'),
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('Second moment of the gradient', 2, None, '___sec69'),
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('RMS prop', 2, None, '___sec70'),
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('ADAM optimizer', 2, None, '___sec71'),
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('Practical tips', 2, None, '___sec72')]}
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end of tocinfo -->
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<body>
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@@ -196,48 +208,57 @@ MathJax.Hub.Config({
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<!-- navigation toc: --> <li><a href="._Splines-bs020.html#___sec19" style="font-size: 80%;">Steepest descent method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs021.html#___sec20" style="font-size: 80%;">Steepest descent method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs022.html#___sec21" style="font-size: 80%;">Final expressions</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs023.html#___sec22" style="font-size: 80%;">Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs024.html#___sec23" style="font-size: 80%;">The routine for the steepest descent method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs025.html#___sec24" style="font-size: 80%;">Steepest descent example</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs026.html#___sec25" style="font-size: 80%;">Conjugate gradient</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs027.html#___sec26" style="font-size: 80%;">Revisiting our first homework</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs028.html#___sec27" style="font-size: 80%;">Gradient descent example</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs029.html#___sec28" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs030.html#___sec29" style="font-size: 80%;">The Hessian matrix</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs031.html#___sec30" style="font-size: 80%;">Simple program</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs032.html#___sec31" style="font-size: 80%;">Gradient Descent Example</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs033.html#___sec32" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs034.html#___sec33" style="font-size: 80%;">Gradient descent and Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs035.html#___sec34" style="font-size: 80%;">Automatic differentiation</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs036.html#___sec35" style="font-size: 80%;">Using autograd</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs037.html#___sec36" style="font-size: 80%;">Autograd with more complicated functions</a></li>
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||||
<!-- navigation toc: --> <li><a href="#___sec37" style="font-size: 80%;">More complicated functions using the elements of their arguments directly</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs039.html#___sec38" style="font-size: 80%;">Functions using mathematical functions from Numpy</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs040.html#___sec39" style="font-size: 80%;">More autograd</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs041.html#___sec40" style="font-size: 80%;">And with loops</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs042.html#___sec41" style="font-size: 80%;">Using recursion</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs043.html#___sec42" style="font-size: 80%;">Unsupported functions</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs044.html#___sec43" style="font-size: 80%;">The syntax a.dot(b) when finding the dot product</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs045.html#___sec44" style="font-size: 80%;">Recommended to avoid</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs046.html#___sec45" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs047.html#___sec46" style="font-size: 80%;">Computation of gradients</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs048.html#___sec47" style="font-size: 80%;">SGD example</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs049.html#___sec48" style="font-size: 80%;">The gradient step</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs050.html#___sec49" style="font-size: 80%;">Simple example code</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs051.html#___sec50" style="font-size: 80%;">When do we stop?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs052.html#___sec51" style="font-size: 80%;">Slightly different approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs053.html#___sec52" style="font-size: 80%;">Program for stochastic gradient</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs054.html#___sec53" style="font-size: 80%;">Momentum based methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs055.html#___sec54" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs023.html#___sec22" style="font-size: 80%;">Code examples for steepest descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs024.html#___sec23" style="font-size: 80%;">Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs025.html#___sec24" style="font-size: 80%;">The routine for the steepest descent method</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs026.html#___sec25" style="font-size: 80%;">Steepest descent example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs027.html#___sec26" style="font-size: 80%;">Conjugate gradient</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs028.html#___sec27" style="font-size: 80%;">Revisiting our first homework</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs029.html#___sec28" style="font-size: 80%;">Gradient descent example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs030.html#___sec29" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs031.html#___sec30" style="font-size: 80%;">The Hessian matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs032.html#___sec31" style="font-size: 80%;">Simple program</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs033.html#___sec32" style="font-size: 80%;">Gradient Descent Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs034.html#___sec33" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs035.html#___sec34" style="font-size: 80%;">Gradient descent and Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs036.html#___sec35" style="font-size: 80%;">Automatic differentiation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs037.html#___sec36" style="font-size: 80%;">Using autograd</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec37" style="font-size: 80%;">Autograd with more complicated functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs039.html#___sec38" style="font-size: 80%;">More complicated functions using the elements of their arguments directly</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs040.html#___sec39" style="font-size: 80%;">Functions using mathematical functions from Numpy</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs041.html#___sec40" style="font-size: 80%;">More autograd</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs042.html#___sec41" style="font-size: 80%;">And with loops</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs043.html#___sec42" style="font-size: 80%;">Using recursion</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs044.html#___sec43" style="font-size: 80%;">Unsupported functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs045.html#___sec44" style="font-size: 80%;">The syntax a.dot(b) when finding the dot product</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs046.html#___sec45" style="font-size: 80%;">Recommended to avoid</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs047.html#___sec46" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs048.html#___sec47" style="font-size: 80%;">Computation of gradients</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs049.html#___sec48" style="font-size: 80%;">SGD example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs050.html#___sec49" style="font-size: 80%;">The gradient step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs051.html#___sec50" style="font-size: 80%;">Simple example code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs052.html#___sec51" style="font-size: 80%;">When do we stop?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs053.html#___sec52" style="font-size: 80%;">Slightly different approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs054.html#___sec53" style="font-size: 80%;">Program for stochastic gradient</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs055.html#___sec54" style="font-size: 80%;">Momentum based methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs056.html#___sec55" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs057.html#___sec56" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs058.html#___sec57" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs059.html#___sec58" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs060.html#___sec59" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs059.html#___sec58" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs060.html#___sec59" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs061.html#___sec60" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs062.html#___sec61" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs063.html#___sec62" style="font-size: 80%;">Simple implementation of the Conjugate gradient algorithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs064.html#___sec63" style="font-size: 80%;">Broyden–Fletcher–Goldfarb–Shanno algorithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs063.html#___sec62" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs064.html#___sec63" style="font-size: 80%;">Simple implementation of the Conjugate gradient algorithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs065.html#___sec64" style="font-size: 80%;">Broyden–Fletcher–Goldfarb–Shanno algorithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs066.html#___sec65" style="font-size: 80%;">Using gradient descent methods, limitations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs067.html#___sec66" style="font-size: 80%;">Momentum based GD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs068.html#___sec67" style="font-size: 80%;">More on momentum based approaches</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs069.html#___sec68" style="font-size: 80%;">Momentum parameter</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs070.html#___sec69" style="font-size: 80%;">Second moment of the gradient</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs071.html#___sec70" style="font-size: 80%;">RMS prop</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs072.html#___sec71" style="font-size: 80%;">ADAM optimizer</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs073.html#___sec72" style="font-size: 80%;">Practical tips</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
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@@ -253,36 +274,52 @@ MathJax.Hub.Config({
|
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<a name="part0038"></a>
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<!-- !split -->
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|
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<h2 id="___sec37" class="anchor">More complicated functions using the elements of their arguments directly </h2>
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<h2 id="___sec37" class="anchor">Autograd with more complicated functions </h2>
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<p>
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To differentiate with respect to two (or more) arguments of a Python
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function, Autograd need to know at which variable the function if
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being differentiated with respect to.
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">f3</span>(x): <span style="color: #408080; font-style: italic"># Assumes x is an array of length 5 or higher</span>
|
||||
<span style="color: #008000; font-weight: bold">return</span> <span style="color: #666666">2*</span>x[<span style="color: #666666">0</span>] <span style="color: #666666">+</span> <span style="color: #666666">3*</span>x[<span style="color: #666666">1</span>] <span style="color: #666666">+</span> <span style="color: #666666">5*</span>x[<span style="color: #666666">2</span>] <span style="color: #666666">+</span> <span style="color: #666666">7*</span>x[<span style="color: #666666">3</span>] <span style="color: #666666">+</span> <span style="color: #666666">11*</span>x[<span style="color: #666666">4</span>]<span style="color: #666666">**2</span>
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">f2</span>(x1,x2):
|
||||
<span style="color: #008000; font-weight: bold">return</span> <span style="color: #666666">3*</span>x1<span style="color: #666666">**3</span> <span style="color: #666666">+</span> x2<span style="color: #666666">*</span>(x1 <span style="color: #666666">-</span> <span style="color: #666666">5</span>) <span style="color: #666666">+</span> <span style="color: #666666">1</span>
|
||||
|
||||
f3_grad <span style="color: #666666">=</span> grad(f3)
|
||||
<span style="color: #408080; font-style: italic"># By sending the argument 0, Autograd will compute the derivative w.r.t the first variable, in this case x1</span>
|
||||
f2_grad_x1 <span style="color: #666666">=</span> grad(f2,<span style="color: #666666">0</span>)
|
||||
|
||||
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">0</span>,<span style="color: #666666">4</span>,<span style="color: #666666">5</span>)
|
||||
<span style="color: #408080; font-style: italic"># ... and differentiate w.r.t x2 by sending 1 as an additional arugment to grad</span>
|
||||
f2_grad_x2 <span style="color: #666666">=</span> grad(f2,<span style="color: #666666">1</span>)
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Print the computed gradient:</span>
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"The computed gradient of f3 is: "</span>, f3_grad(x))
|
||||
x1 <span style="color: #666666">=</span> <span style="color: #666666">1.0</span>
|
||||
x2 <span style="color: #666666">=</span> <span style="color: #666666">3.0</span>
|
||||
|
||||
<span style="color: #408080; font-style: italic"># The analytical gradient is: (2, 3, 5, 7, 22*x[4])</span>
|
||||
f3_grad_analytical <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([<span style="color: #666666">2</span>, <span style="color: #666666">3</span>, <span style="color: #666666">5</span>, <span style="color: #666666">7</span>, <span style="color: #666666">22*</span>x[<span style="color: #666666">4</span>]])
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"Evaluating at x1 = </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">, x2 = </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">"</span><span style="color: #666666">%</span>(x1,x2))
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"-"</span><span style="color: #666666">*30</span>)
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Print the analytical gradient:</span>
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"The analytical gradient of f3 is: "</span>, f3_grad_analytical)
|
||||
<span style="color: #408080; font-style: italic"># Compare with the analytical derivatives:</span>
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Derivative of f2 w.r.t x1 is: 9*x1**2 + x2:</span>
|
||||
f2_grad_x1_analytical <span style="color: #666666">=</span> <span style="color: #666666">9*</span>x1<span style="color: #666666">**2</span> <span style="color: #666666">+</span> x2
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Derivative of f2 w.r.t x2 is: x1 - 5:</span>
|
||||
f2_grad_x2_analytical <span style="color: #666666">=</span> x1 <span style="color: #666666">-</span> <span style="color: #666666">5</span>
|
||||
|
||||
<span style="color: #408080; font-style: italic"># See the evaluated derivations:</span>
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"The derivative of f2 w.r.t x1: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">"</span><span style="color: #666666">%</span>( f2_grad_x1(x1,x2) ))
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"The analytical derivative of f2 w.r.t x1: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">"</span><span style="color: #666666">%</span>( f2_grad_x1(x1,x2) ))
|
||||
|
||||
<span style="color: #008000; font-weight: bold">print</span>()
|
||||
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"The derivative of f2 w.r.t x2: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">"</span><span style="color: #666666">%</span>( f2_grad_x2(x1,x2) ))
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"The analytical derivative of f2 w.r.t x2: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">"</span><span style="color: #666666">%</span>( f2_grad_x2(x1,x2) ))
|
||||
</pre></div>
|
||||
<p>
|
||||
Note that in this case, when sending an array as input argument, the
|
||||
output from Autograd is another array. This is the true gradient of
|
||||
the function, as opposed to the function in the previous example. By
|
||||
using arrays to represent the variables, the output from Autograd
|
||||
might be easier to work with, as the output is closer to what one
|
||||
could expect form a gradient-evaluting function.
|
||||
Note that the grad function will not produce the true gradient of the function. The true gradient of a function with two or more variables will produce a vector, where each element is the function differentiated w.r.t a variable.
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -310,7 +347,7 @@ could expect form a gradient-evaluting function.
|
||||
<li><a href="._Splines-bs046.html">47</a></li>
|
||||
<li><a href="._Splines-bs047.html">48</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._Splines-bs064.html">65</a></li>
|
||||
<li><a href="._Splines-bs073.html">74</a></li>
|
||||
<li><a href="._Splines-bs039.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
Reference in New Issue
Block a user